1 Mean-Squared Error Beamforming for Signal Estimation: A ...nehorai/paper/arra_chapter_final2.pdf · of a narrow-band signal. The source signal amplitude s(t) may be a deterministic

1 Mean-Squared ErrorBeamforming for SignalEstimation: A CompetitiveApproach

YONINA C. ELDAR and ARYE NEHORAI

1.1 INTRODUCTION

Beamforming is a classical method of processing temporal sensor array measure-

ments for signal estimation, interference cancellation, source direction, and spectrum

estimation. It has ubiquitously been applied in areas such as radar, sonar, wire-

less communications, speech processing, and medical imaging (see, for example,

[1, 2, 3, 4, 5, 6, 7, 8] and the references therein.)

Conventional approaches for designing data dependent beamformers typically

attempt to maximize the signal-to-interference-plus-noise ratio (SINR). Maximizing

the SINR requires knowledge of the interference-plus-noise covariance matrix and

the array steering vector. Since this covariance is unknown, it is often replaced by

the sample covariance of the measurements, resulting in deterioration of performance

with higher signal-to-noise ratio (SNR) when the signal is present in the training data.

Some beamforming techniques are designed to mitigate this effect [9, 10, 11, 12],

whereas others are developed to also overcome uncertainty in the steering vector,

for example [13, 14, 15, 16, 17, 18, 19] (see also Chapters 2 and 3 in this book

and the references therein.) Despite the fact that the SINR has been used as a

measure of beamforming performance and as a design criterion in many beamforming

approaches, we note that maximizing SINR may not guarantee a good estimate of

D R A F T September 23, 2004, 8:12pm D R A F T

ii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

the signal. In anestimationcontext, where our goal is to design a beamformer in

order to obtain an estimate of the signal amplitude that is close to its true value, it

would make more sense to choose the weights to minimize an objective that is related

to the estimation error,i.e., the difference between the true signal amplitude and its

estimate, rather than the SINR. Furthermore, in comparing performance of different

beamforming methods, it may be more informative to consider the estimation error

as a measure of performance.

In this chapter we derive five beamformers for estimating a signal in the presence

of interference and noise using the mean-squared error (MSE) as the performance

criterion assuming either known steering vectors or random steering vectors with

known second-order statistics. Computing the MSE shows, however, that it depends

explicitly on the unknown signal magnitude in the deterministic case, or the unknown

signal second order moment in the stochastic case [8], hence cannot be minimized

directly. Thus, we aim at designing a robust beamformer whose performance in terms

of MSE is good across all possible values of the unknowns. To develop a beamformer

with this property, we rely on a recent minimax estimation framework that has been

developed for solving robust estimation problems [20, 21]. This framework considers

a general linear estimation problem, and suggests two classes of linear estimators that

optimize an MSE-based criterion. In the first approach, developed in [20], a linear

estimator is developed to minimize the worst-case MSE over an ellipsoidal region of

uncertainty on the parameter set. In the second approach, developed in [21], a linear

estimator is designed whose performance is as close as possible to that of the optimal

linear estimator for the case of known model parameters. Specifically, the estimator

is designed to minimize the worst-caseregret, which is the difference between the

MSE of the estimator in the presence of uncertainties, and the smallest attainable

MSE with a linear estimator that knows the exact model. Note that as we explain

further in Section 1.2.2, even when the signal magnitude is known, we cannot achieve

a zero MSE with alinear estimator.

Using the general minimax MSE framework [20, 21] we develop two classes of

robust beamformers: Minimax MSE beamformers and minimax regret beamformers.

The minimax MSE beamformers minimize the worst-case MSE over all signals whose


INTRODUCTION iii

magnitude (or variance, in the zero-mean stochastic signals case) is bounded by a

constant. The minimax regret beamformers minimize the worst-case regret over all

bounded signals, where this approach considers both an upper and a lower bound on

the signal magnitude. We note that in practice, if bounds on the signal magnitude

are not known, then they can be estimated from the data, as we demonstrate in the

numerical examples.

We first consider the case in which the steering vector is known completely

and develop a minimax MSE and minimax regret beamformer. In this case we

show that the minimax beamformers are scaled versions of the classical SINR-

based beamformers. We then consider the case in which the steering vector is not

completely known or fully calibrated, for example due to errors in sensor positions,

gains or phases, coherent and incoherent local scatters, receiver fluctuations due

to temperature changes, quantization effects, etc. (see [22, 14] and the references

therein). To model the uncertainties in the steering vector, we assume that it is a

random vector with known mean and covariance. Under this model, we develop three

possible beamformers: minimax MSE, minimax regret, and, following the ideas in

[23], a least-squares beamformer. While the minimax beamformers require bounds

on the signal magnitude, the least-squares beamformer does not require such bounds.

As we show, in the case of a random steering vector, the beamformers resulting from

the minimax approaches are fundamentally different than those resulting from the

SINR approach, and are not just scaled versions of each other as in the known steering

vector case.

To illustrate the advantages of our methods we present several numerical examples

comparing the proposed methods with conventional SINR-based methods, and several

recently proposed robust methods. For a known steering vector, the minimax MSE

and minimax regret beamformers are shown to consistently have the best performance,

particularly for negative SNR values. For random steering vectors, the minimax and

least-squares beamformers are shown to have the best performance for low SNR

values (-10 to 5 dB). As we show, the least-squares approach, which does not require

bounds on the signal magnitude, often performs better than the recently proposed

robust methods [14, 15, 16] for dealing with steering vector uncertainty. In this case,


iv MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

the improvement in performance resulting from the proposed methods is often quite

substantial.

The chapter is organized as follows. In Section 1.2 we present the problem for-

mulation and review existing methods. In Section 1.3 we develop the minimax MSE

and minimax regret beamformers for the case in which the steering vector is known.

The case of a random steering vector is considered in Section 1.4. In Sections 1.5 and

1.6 we discuss practical considerations and present numerical examples illustrating

the advantages of the proposed beamformers over several existing standard and ro-

bust beamformers, for a wide range of SNR values. The chapter is summarized in

Section 1.7.

1.2 BACKGROUND AND PROBLEM FORMULATION

We denote vectors inCM by boldface lowercase letters and matrices inCN×M by

boldface uppercase letters. The matrixI denotes the identity matrix of the appropriate

dimension,(·)∗ denotes the Hermitian conjugate of the corresponding matrix, and

(·) denotes an estimated variable. The eigenvector of a matrixA associated with the

largest eigenvalue is denoted byP {A}.

1.2.1 Background

Beamforming methods are used extensively in a variety of areas, where one of their

goals is to estimate the source signal amplitudes(t) from the array observations

y(t) = s(t)a + i(t) + e(t), 1 ≤ t ≤ N, (1.1)

wherey(t) ∈ CM is the complex vector of array observations at timet with M being

the number of array sensors,s(t) is the signal amplitude,a is the signal steering

vector,i(t) is the interference,e(t) is a Gaussian noise vector andN is the number of

snapshots [4, 6, 8]. In the above model we implicitly made the common assumption

of a narrow-band signal.

The source signal amplitudes(t) may be a deterministic unknown signal, such as

a complex sinusoid, or a stochastic stationary process with unknown signal power.


BACKGROUND AND PROBLEM FORMULATION v

For concreteness, in our development below we will treats(t) as a deterministic

signal. However, as we show analytically in Section 1.2.2 and through simulations

in Section 1.6, the optimality properties of the algorithms we develop are valid also

in the case of stochastic signals.

In some applications, such as in the case of a fully calibrated array, the steering

vector can be assumed to be known exactly. In this case, we treata as a known deter-

ministic vector. However, in practice the array response may have some uncertainties

or perturbations in the steering vectors. These perturbations may be due to errors in

sensor positions, gains or phases, mutual couplings between sensors, receiver fluctu-

ations due to temperature changes, quantization effects, and coherent and incoherent

local scatters [22, 14]. To account for these uncertainties, several authors have tried

modeling some of their effects [24, 25]. However these perturbations often take place

simultaneously, which significantly complicates the model. Instead, the uncertainty

in a can be taken into account by treating it as a deterministic vector that lies in

an ellipsoid centered at a nominal steering vector [14, 15]. An alternative approach

has been to treat the steering vector as a random vector assuming knowledge of its

distribution [13] or the second-order statistics [26, 27, 28, 29, 30, 31, 32, 16]. In

the latter case the mean value ofa corresponds to the nominal steering vector, and

the covariance matrix captures its perturbations. In this chapter we consider the

cases wherea is known (Section 1.3) or random with known second-order statistics

(Section 1.4).

Our goal is to estimate the signal amplitudes(t) from the observationsy(t) using

a set of beamformer weightsw(t), where the output of a narrowband beamformer is

given by

s(t) = w∗(t)y(t), 1 ≤ t ≤ N. (1.2)

To illustrate our approach, in this section we focus primarily on the case in which the

steering vectora is assumed to be known exactly. As we show in Section 1.4, the

essential ideas we outline for this case can also be applied to the case in whicha is

random.

Traditionally, the beamformer weightsw(t) = w (where we omitted the indext

for brevity) are chosen to maximize the SINR, which in the case of a known steering


vi MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

vector is given by

SINR ∝ |w∗a|2w∗Rw

, (1.3)

where

R = E {(i + n)(i + n)∗} (1.4)

is the interference-plus-noise covariance matrix. The weight vector maximizing the

SINR is given by

wMVDR =1

a∗R−1aR−1a. (1.5)

The solution (1.5) is commonly referred to as the minimum variance distortionless

response (MVDR) beamformer, since it can also be obtained as the solution to

minw

w∗Rw subject to w∗a = 1. (1.6)

In practice, the interference-plus-noise covariance matrixR is often not available.

In such cases, the exact covarianceR in (1.5) is replaced by an estimated covariance.

Various methods exist for estimating the covarianceR. The simplest approach is to

choose the estimate as the sample covariance

Rsm =1N

N∑t=1

y(t)y(t)∗. (1.7)

The resulting beamformer is referred to as the sample matrix inversion (SMI) beam-

former or the Capon beamformer [33, 34]. (For simplicity, we assume here that

the sample covariance matrix is invertible.) If the signal is present in the training

data, then it is well known that the performance of the MVDR beamformer withR

replaced byRsm of (1.7) degrades considerably [11].

An alternative approach for estimatingR is the diagonal loading approach, in

which the estimate is chosen as

Rdl = Rsm + ξI =1N

N∑t=1

y(t)y(t)∗ + ξI, (1.8)

whereξ is the diagonal loading factor. The resulting beamformer is referred to as the

loaded SMI or the loaded Capon beamformer [9, 10]. Various methods have been

proposed for choosing the diagonal loading factorξ; seee.g.,[10]. A heuristic choice


BACKGROUND AND PROBLEM FORMULATION vii

for ξ, which is common in applications, isξ ≈ 10σ2, whereσ2 is the noise power in

a single sensor.

Another popular approach to estimatingR is the eigenspace approach [11], in

which the inverse of the covariance matrix is estimated as

(Reig

)−1

=(Rsm

)−1

Ps, (1.9)

wherePs is the orthogonal projection onto the sample signal+interference subspace,

i.e., the subspace corresponding to theD + 1 largest eigenvalues ofRsm, wereD is

the known rank of the interference subspace.

Note that the Capon, loaded Capon and eigenspace beamformers, can all be viewed

as MVDR beamformers with a particular estimate ofR.

The class of MVDR beamformers assumes explicitly that the steering vectora is

known exactly. Recently, several robust beamformers have been proposed for the case

in which the steering vector is not known precisely, but rather lies in some uncertainty

set [14, 15, 16]. Although originally developed to deal with steering vector mismatch,

the authors of the referenced papers suggest using these robust methods even in the

case in whicha is known, in order to deal with the mismatch in the interference-

plus-noise covariance, namely the finite sample effects, and the fact that the signal is

typically present in the training data. Each of the above robust methods is designed to

maximize a measure of SINR on the uncertainty set. Specifically, in [14], the authors

suggest minimizingw∗Rsmw subject to the constraint that|w∗c| ≥ 1 for all possible

values of the steering vectorc, where‖c − a‖ ≤ ε. The resulting beamformer is

given by

w =λ

λa∗(Rsm + λε2I

)−1

a− 1

(Rsm + λε2I

)−1

a, (1.10)

whereλ is chosen such that|w∗a − 1|2 = ε2w∗w. In practice, the solution can be

found by using a second order cone program. In [15] the authors consider a similar

approach in which they first estimate the steering vector by minimizinga∗R−1sma

with respect toa subject to‖a − a‖2 = ε, and then use√

M a/‖a‖ in the MVDR


viii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

beamformer, which results in the beamformer

w =α√M

·

(λRsm + I

)−1

a

a∗(λRsm + I

)−1

Rsm

(λRsm + I

)−1

a. (1.11)

Hereλ is chosen such that

∥∥∥∥(I + λRsm

)−1

a∥∥∥∥

2

= ε, andα =∥∥∥∥(R−1

sm + λI)−1

a∥∥∥∥.

Finally, in [16] the authors consider a general-rank signal model. Adapting their

results to the rank-one steering vector case, their beamformer is the solution to

minimizingw∗Rdlw subject to|w∗a|2 ≥ 1−w∗∆w for all ‖∆‖ ≤ ε, and is given

by

w = αP{R−1

dl (aa∗ − εI)}

, (1.12)

whereα is chosen such thatw∗(aa∗ − εI)w = 1.

The motivation behind the class of MVDR beamformers and the robust beam-

formers is to maximize the SINR. However, choosingw to maximize the SINR does

not necessarily result in an estimated signal amplitudes(t) that is close tos(t). In an

estimationcontext, where our goal is to design a beamformer in order to obtain an

estimates(t) that is close tos(t), it would make more sense to choose the weightsw

to minimize the MSE rather than to maximize the SINR, which is not directly related

to the estimation errors(t)− s(t).

1.2.2 MSE Beamforming

If s = w∗y, then, assuming thats is deterministic, the MSE betweens ands is given

by

E{|s− s|2} = V (s) + |B(s)|2 = w∗Rw + |s|2|1−w∗a|2, (1.13)

whereV (s) = E{|s− E {s} |2} is the variance of the estimates and B(s) =

E {s} − s is the bias. In the case in whichs is a zero-mean random variable with

varianceσ2s , the MSE is given by

E{|s− s|2} = w∗Rw + σ2

s |1−w∗a|2. (1.14)


BACKGROUND AND PROBLEM FORMULATION ix

Comparing (1.13) and (1.14) we see that the expressions for the MSE have the same

form in the deterministic and stochastic cases, where|s|2 in the deterministic case is

replaced byσ2s in the stochastic case. For concreteness, in the discussion in the rest

of the chapter we assume the deterministic model. However, all the results hold true

for the stochastic model where we replace|s|2 everywhere withσ2s . In particular, in

the development of the minimax MSE and regret beamformers in the stochastic case,

the bounds on|s|2 are replaced by bounds on the signal varianceσ2s .

The minimum MSE (MMSE) beamformer minimizing the MSE when|s| is known

is obtained by differentiating (1.13) with respect tow and equating to0, which results

in

w(s) = |s|2(R + |s|2aa∗)−1a. (1.15)

Using the Matrix Inversion Lemma we can expressw(s) as

w(s) =|s|2

1 + |s|2a∗R−1aR−1a. (1.16)

The MMSE beamformer can alternatively be expressed as

w(s) =|s|2a∗R−1a

1 + |s|2a∗R−1a· R−1aa∗R−1a

= β(s)wMVDR, (1.17)

where

β(s) =|s|2a∗R−1a

1 + |s|2a∗R−1a. (1.18)

The scalingβ(s) satisfies0 ≤ β(s) ≤ 1, and is monotonically increasing in|s|2.

Therefore, for any|s|2, ‖w(s)‖ ≤ ‖wMVDR‖. Substitutingw(s) back into (1.13),

the smallest possible MSE, which we denote byMSEOPT, is given by

MSEOPT =|s|2

1 + |s|2a∗R−1a. (1.19)

Using (1.13) we can also compute the MSE of the MVDR beamformer (1.5),

which maximizes the SINR, assuming thatR is known. Substitutingw =

(1/a∗R−1a)R−1a into (1.13), the MSE is

MSEMVDR =1

a∗R−1a. (1.20)


x MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

Comparing (1.19) with (1.20),

MSEOPT =|s|2

1 + |s|2a∗R−1a=

11/|s|2 + a∗R−1a

≤ 1a∗R−1a

= MSEMVDR.

(1.21)

Thus, as we expect, the MMSE beamformer always results in a smaller MSE than

the MVDR beamformer. Therefore, in an estimation context where our goal is to

estimate the signal amplitude, the MMSE beamformer will lead to better average

performance.

From (1.17) we see that the MMSE beamformer is just a shrinkage of the MVDR

beamformer (as we will show below, this is no longer true in the case of a random

steering vector). Therefore, the two beamformers will result in the same SINR, so

that the MMSE beamformer also maximizes the SINR. However, as (1.21) shows, this

shrinkage factor impacts the MSE so that the MMSE beamformer has better MSE

performance. To illustrate the advantage of the MMSE beamformer, we consider

a numerical example. The scenario consists of a uniform linear array (ULA) of

M = 20 omnidirectional sensors spaced half a wavelength apart. We chooses as a

complex sinewave with varying amplitude to obtain the desired SNR in each sensor;

its plane-wave has a DOA of30o relative to the array normal. The noisee is a

zero-mean, Gaussian, complex random vector, temporally and spatially white, with

a power of 0 dB in each sensor. The interference is given byi = aii wherei is

a zero-mean, Gaussian, complex process temporally white with interference-plus-

noise ratio (INR) of 20 dB andai is the interference steering vector with DOA =

−30o. Assuming knowledge of|s|2, R anda, we evaluate the square-root of the

normalized MSE (NMSE) over a time-window ofN = 100 samples, where each

result is obtained by averaging 200 Monte Carlo simulations.

Figure 1.1 illustrates the NMSE of the MMSE and MVDR beamformers when

estimating the complex sinewave, as a function of SNR. It can be seen that the MMSE

beamformer outperforms the MVDR beamformer in all the illustrated SNR range.

As the SNR increases, the MVDR beamformer converges to the MMSE beamformer,

sinceβ(s) in (1.17) converges to1. The absolute value of the original signal and

its estimates obtained from the MMSE and MVDR beamformers are illustrated in


BACKGROUND AND PROBLEM FORMULATION xi

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNR [dB]

Squ

are

Roo

t of N

orm

aliz

ed M

SE

MMSEMVDR

Fig. 1.1 Square-root of the normalized MSE as a function of SNR using the MMSE and

MVDR beamformers when estimating a complex sinewave with DOA30o, in the presence of

an interference with INR = 20 dB and DOA =−30o. We assume that|s|2, the noise-plus-

interference covariance matrixR and the steering vectora are known.

Fig. 1.2, for an SNR of−10 dB. Clearly, the MMSE beamformer leads to a better

estimate than the MVDR beamformer.

The difference between the MSE and SINR based approaches is more pronounced

in the case of a random steering vector. Suppose thata is a random vector with mean

m and covariance matrixC. In Section 1.4 we consider this case in detail, and show

that the beamformer minimizing the MSE is

w(s) =|s|2

1 + |s|2m∗ (R + |s|2C)−1 m

(R + |s|2C)−1

m. (1.22)

On the other hand, the beamformer maximizing the SINR is given by

w = αP {R−1(C + mm∗)

}, (1.23)

whereα is chosen such thatw∗(C + mm∗)w = 1. We refer to this beamformer

as the principal eigenvector (PEIG) beamformer [35]. Comparing (1.22) and (1.23)

we see that in this case the two beamformers are in general not scalar versions of


xii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of sample

Abs

olut

e va

lue

Signal magnitudeMMSE estimate MVDR estimate

Fig. 1.2 Absolute values of the true complex sinewave, and its estimates obtained by the

MMSE and MVDR beamformers for SNR = -10 dB, in the presence of an interference with

INR = 20 dB and DOA =−30o. We assume that|s|2, the noise-plus-interference covariance

matrixR and the steering vectora are known.

each other. As we now illustrate, the MMSE beamformer can result in a much

better estimate of the signal amplitude and waveform than the SINR-based PEIG

beamformer.

To illustrate the advantage of the MMSE beamformer in the case of a random

steering vector, we consider an example in which the DOA of the signal is random.

The scenario is similar to that of the previous example, where in place of a constant

signal DOA, the DOA is now given by a Gaussian random variable with mean equal

to 30o and standard deviation equal to 1 (about±3o). The meanm and covariance

matrix C of the steering vector are estimated from 2000 realizations of the steering

vector. Assuming knowledge of|s|2 andR, we evaluate the NMSE over a time-

window ofN = 100 samples, where each result is obtained by averaging 200 Monte

Carlo simulations.

Figure 1.3 illustrates the NMSE of the MMSE and PEIG beamformers as a function

of SNR. It can be seen that the NMSE of the MMSE beamformer is substantially

lower than that of the PEIG beamformer. The absolute value of the original signal


BACKGROUND AND PROBLEM FORMULATION xiii

−10 −8 −6 −4 −2 0 2 40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR [dB]

Squ

are

root

of N

MS

EMMSEPEIG

Fig. 1.3 Square-root of the normalized MSE as a function of SNR using the MMSE and

PEIG beamformers when estimating a complex sinewave with random DOA, in the presence

of an interference with INR = 20 dB and DOA =−30o. We assume that|s|2, the noise-plus-

interference covariance matrixR, and the meanm and covarianceC of the steering vectora

are known.

and its estimates obtained from the MMSE and PEIG beamformers are illustrated in

Fig. 1.4 for an SNR of−10 dB. Clearly, the MMSE beamformer leads to a better

signal estimate than the PEIG beamformer. It is also evident from this example

that the signal waveform estimate obtained from both beamformers is different: the

MMSE beamformer leads to a signal waveform that is much closer to the original

waveform than the PEIG beamformer.

1.2.3 Robust MMSE Beamforming

Unfortunately, both in the case of knowna and in the case of randoma the MMSE

beamformer depends explicitly on|s| which is typically unknown. Therefore, in

practice, we cannot implement the MMSE beamformer. The problem stems from

the fact that the MSE depends explicitly on|s|. To illustrate the main ideas, in the

remainder of this section we focus on the case of a deterministic steering vector.


xiv MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of sample

Abs

olut

e va

lue

Signal magnitudeMMSE estimate PEIG estimate

Fig. 1.4 Absolute values of the true complex sinewave with random DOA, and its estimates

obtained by the MMSE and PEIG beamformers for SNR = -10 dB, in the presence of an

interference with INR = 20 dB and DOA =−30o. We assume that|s|2, the noise-plus-

interference covariance matrixR, and the meanm and covarianceC of the steering vectora

are known.


BACKGROUND AND PROBLEM FORMULATION xv

One approach to obtain a beamformer that does not depend on|s| in this case is to

force the term depending on|s|, namely the bias, to0, and then minimize the MSE,

i.e.,

minw

w∗Rw subject to w∗a = 1, (1.24)

which leads to the class of MVDR beamformers. Thus, in addition to maximizing

the SINR, the MVDR beamformer minimizes the MSE subject to the constraint that

the bias in the estimators is equal to0. However, this does not guarantee a small

MSE, so that on average, the resulting estimate ofs may be far froms. Indeed, it

is well known that unbiased estimators may often lead to large MSE values. The

attractiveness of the SINR criterion is the fact that it is easy to solve, and leads to a

beamformer that does not depend on|s|. Its drawback is that it does not necessarily

lead to a small MSE, as can also be seen in Figs. 1.1 and 1.3.

We note, that as we discuss in Section 1.4.1, the property of the MVDR beamformer

that it minimizes the MSE subject to a zero bias constrain no longer holds in the

random steering vector case. In fact, whena is random with positive covariance

matrix, there is in general no choice of linear beamformer for which the MSE is

independent of|s|.Instead of forcing the term depending on|s| to zero, it would be desirable to design

a robust beamformer whose MSE is reasonably small across all possible values of

|s|. To this end we need to define the set of possible values of|s|. In some practical

applications, we may knowa-priori bounds on|s|, for example when the type of the

source and the possible distances from the array are known, as can happen for instance

in wireless communications and underwater source localization. We may have an

upper bound of the form|s| ≤ U , or we may have both an upper and (nonzero) lower

bound, so that

L ≤ |s| ≤ U. (1.25)

In our development of the minimax robust beamformers, we will assume that the

boundsL andU are known. In practice, if no such bounds are knowna-priori, then

we can estimate them from the data, as we elaborate on further in Sections 1.5 and

1.6. This is similar in spirit to the MVDR-based beamformers: In developing the


xvi MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

MVDR beamformer it is assumed that the interference-plus-noise covariance matrix

R is known; however, in practice, this matrix is estimated from the data.

Given an uncertainty set of the form (1.25), we may seek a beamformer that

minimizes a worst-case MSE measure on this set. In the next section, we rely on

ideas of [21] and [20], and propose two robust beamformers. We first assume that

only an upper bound on|s| is given, and develop a minimax MSE beamformer

that minimizes the worst-case MSE over all|s| ≤ U . As we show in (1.34), this

beamformer is also minimax subject to (1.25). We then develop a minimax regret

beamformer over the set defined by (1.25), that minimizes the worst-case difference

between the MSE attainable with a beamformer that does not know|s|, and the

optimal MSE of the MMSE beamformer that minimizes the MSE when|s| is known.

In Section 1.4 we develop minimax MSE and minimax regret beamformers for the

case of a random steering vector.

1.3 MINIMAX MSE BEAMFORMING FOR KNOWN STEERING

VECTOR

We now consider two MSE-based criteria for developing robust beamformers when

the steering vector is known: In Section 1.3.1 we consider a minimax MSE approach

and in Section 1.3.2 we consider a minimax regret approach. In our development,

we assume that the covariance matrixR is known. In practice, as we discuss in

Sections 1.5 and 1.6, the unknownR is replaced by an estimateR.

1.3.1 Minimax MSE Beamforming

The first approach we consider for developing a robust beamformer it to minimize

the worst-case MSE over all bounded values of|s|. Thus, we seek the beamformer

that is the solution to

minw

max|s|≤U

E{|s− s|2} = min

wmax|s|≤U

{w∗Rw + |s|2|1−w∗a|2} . (1.26)

This ensures that in the worst case, the MSE of the resulting beamformer is minimized.


MINIMAX MSE BEAMFORMING FOR KNOWN STEERING VECTOR xvii

Now,

max|s|≤U

{w∗Rw + |s|2|1−w∗a|2} = w∗Rw + U2|1−w∗a|2, (1.27)

which is equal to the MSE of (1.13) with|s|2 = U . It follows that the minimax MSE

beamformer, denotedwMXM, is an MMSE beamformer of the form (1.16) matched

to |s| = U :

wMXM =U2

1 + U2a∗R−1aR−1a = βMXMwMVDR, (1.28)

where

βMXM =U2a∗R−1a

1 + U2a∗R−1a. (1.29)

The resulting MSE is

MSEMXM =U4a∗R−1a + |s|2(1 + U2a∗R−1a)2

. (1.30)

For any|s|2 ≤ U2, we have that

MSEMXM ≤ U2

1 + U2a∗R−1a. (1.31)

For comparison, we have seen in (1.20) that the MSE of the MVDR beamformer is

MSEMVDR =1

a∗R−1a, (1.32)

from which we have immediately that for any choice ofU ,

MSEMVDR > MSEMXM. (1.33)

The inequality (1.33) is valid as long as the true covarianceR is known. When

R is estimated from the data, (1.33) is no longer true in general. Nonetheless, as we

will see in the simulations in Section 1.6, (1.33) typically holds even when bothU

andR are estimated from the data.

Finally, we point out that the minimax MSE beamformerwMXM of (1.30) is also

the solution in the case in which we have both an upper and lower bound on the norm


xviii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

of s. This follows from the fact that

maxL≤|s|≤U

|s|2|1−w∗a|2 = max|s|≤U

|s|2|1−w∗a|2, (1.34)

since the maximum is obtained at|s| = U . Thus, in contrast with the minimax regret

beamformer developed in the next section, the minimax MSE beamformer does not

take the lower bound (if available) into account and therefore may be overconservative

when such a lower bound is known.

1.3.2 Minimax Regret Beamforming

We have seen that the minimax MSE beamformer is an MMSE beamformer matched

to the worst possible choice of|s|, namely|s| = U . In some practical applications,

particulary when a lower bound on|s| is known, this approach may be overcon-

servative. Although it optimizes the performance in the worst case, it may lead to

deteriorated performance in other cases. To overcome this possible limitation, in this

section we develop a minimax regret beamformer whose performance is as close as

possible to that of the MMSE beamformer that knowss, for all possible values ofs

in a prespecified region of uncertainty. Thus, we ensure that over a wide range of

values ofs, our beamformer will result in a relatively low MSE.

In [21], a minimax difference regret estimator was derived for the problem of

estimating an unknown vectorx in a linear modely = Hx+n, whereH is a known

linear transformation, andn is a noise vector with known covariance matrix. The

estimator was designed to minimize the worst case regret over all bounded vectorsx,

namely vectors satisfyingx∗Tx ≤ U2 for someU > 0 and positive definite matrix

T. It was shown that the linear minimax regret estimator can be found as a solution

to a convex optimization problem that can be solved very efficiently.

In our problem, the unknown parameterx = s is a scalar, so that an explicit

solution can be derived, as we show below (see also [36]). Furthermore, in our

development we consider both lower and upper bounds on|s|, so that we seek the

beamformer that minimizes the worst case regret over the uncertainty region (1.25).

The minimax regret beamformerwMXR is designed to minimize the worst-case

regret subject to the constraintL ≤ |s| ≤ U , where the regret, denotedR(s,w),


MINIMAX MSE BEAMFORMING FOR KNOWN STEERING VECTOR xix

MSE

Known |s|

Regret

|s|

Unknown |s|

Fig. 1.5 The solid line represents the best attainable MSE as a function of|s| when |s| is

known. The dashed line represents a desirable graph of MSE with small regret as a function

of |s| using some linear estimator that does not depend on|s|.

is defined as the difference between the MSE using an estimators = w∗y and the

smallest possible MSE attainable with an estimator of the forms = w∗(s)y when

s is known, so thatw can depend explicitly ons. We have seen in (1.19) that since

we are restricting ourselves to linear beamformers, even in the case in which the

beamformer can depend ons, the minimal attainable MSE is not generally equal

to zero. The best possible MSE is illustrated schematically in Fig. 1.5. Instead of

seeking an estimator to minimize the worst-case MSE, we therefore propose seeking

an estimator to minimize the worst-case difference between its MSE and the best

possible MSE, as illustrated in Fig. 1.5.

Using (1.19), we can express the regret as

R(s,w) = E{|w∗y − s|2}−MSEOPT

= w∗Rw + |s|2|1−w∗a|2 − |s|21 + |s|2a∗R−1a

. (1.35)


xx MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

ThuswMXR is the solution to

minw

maxL≤|s|≤U

R(s,w) =

= minw

{w∗Rw + max

L≤|s|≤U

{|s|2|1−w∗a|2 − |s|2

1 + |s|2a∗R−1a

}}. (1.36)

The minimax regret beamformer is given by the following theorem.

Theorem 1. Let s denote an unknown signal amplitude in the modely = sa + n,

wherea is a known length-M vector, andn is a zero-mean random vector with

covarianceR. Then the solution to the problem

mins=w∗y

maxL≤|s|≤U

{E

{|s− s|2}− mins=w∗(s)y

E{|s− s|2}

}=

= minw maxL≤|s|≤U

{w∗Rw + |s|2(1−w∗a)2 − |s|2

1+|s|2a∗R−1a

}

is

s =

(1− 1√

(1 + L2a∗R−1a)(1 + U2a∗R−1a)

)a∗R−1ya∗R−1a

.

Before proving Theorem 1, we first comment on some of the properties of the

minimax regret beamformer, which, from Theorem 1, is given by

wMXR =

(1− 1√

(1 + L2a∗R−1a)(1 + U2a∗R−1a)

)1

a∗R−1aR−1a. (1.37)

Comparing (1.37) with (1.5) we see that the minimax regret beamformer is a scaled

version of the MVDR beamformer, i.e.wMXR = βMXRwMVDR where

βMXR = 1− 1√(1 + L2a∗R−1a)(1 + U2a∗R−1a)

. (1.38)

Clearly, the scalingβMXR satisfies0 ≤ βMXR ≤ 1, and is monotonically increasing

in L andU . In addition, whenU = L,

βMXR = 1− 11 + U2a∗R−1a2

=U2a∗R−1a

1 + U2a∗R−1a, (1.39)

in which case the minimax regret beamformer is equal to the minimax MSE beam-

former of (1.30). ForL < U , ‖wMXR‖ < ‖wMXM‖.


MINIMAX MSE BEAMFORMING FOR KNOWN STEERING VECTOR xxi

It is also interesting to note that the minimax regret beamformer can be viewed as

an MMSE beamformer of the form (1.16), matched to

|s|2 =1

a∗R−1a

(√(1 + L2a∗R−1a)(1 + U2a∗R−1a)− 1

), (1.40)

for arbitrary choices ofL andU . This follows immediately from substituting|s|2 of

(1.40) into (1.16). Since the minimax regret estimator minimizes the MSE for the

signal power given by (1.40), we may view this power as the “least-favorable” signal

power in the regret sense.

The signal power (1.40) can be viewed as an estimate of the true, unknown signal

power. To gain some insight into the estimate of (1.40) we note that

(√(1 + L2γ)(1 + U2γ)− 1

)(√1 + L2γ +

√1 + U2γ

)=

= U2γ√

1 + L2γ + L2γ√

1 + U2γ, (1.41)

where for brevity, we denotedγ = a∗R−1a. Substituting (1.41) into (1.40), we have

that

|s|2 =U2√

1 + L2a∗R−1a + L2√

1 + U2a∗R−1a√1 + L2a∗R−1a +

√1 + U2a∗R−1a

. (1.42)

From (1.42) it follows that the unknown signal power is estimated as a weighted

combination of the power boundsU2 andL2, where the weights depend explicitly

on the uncertainty set and onµ(L) andµ(U), where

µ(T ) = T 2a∗R−1a, (1.43)

can be viewed as the SNR in the observations, when the signal power isT 2.

If µ(L) À 1, then from (1.40),

|s|2 ≈√

U2L2, (1.44)

so that in this case the unknown signal power is estimated as the geometric mean of

the power bounds. If, on the other hand,µ(L) ¿ 1, then from (1.42),

|s|2 ≈ 12

(L2 + U2

). (1.45)


xxii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

Thus, in this case the unknown signal power is estimated as the algebraic mean of

the power bounds.

It is interesting to note that while the minimax MSE estimator of (1.28) is matched

to a signal powerU2, the minimax difference regret estimator of (1.37) is matched to

a signal power|s|2 ≤ (U2 +L2)/2. This follows from (1.40) by using the inequality√

ab ≤ (a + b)/2.

We now prove Theorem 1. Although parts of the proof are similar to the proofs in

[37], we repeat the arguments for completeness.

Proof. To develop a solution to (1.36), we first consider the inner maximization

problem

f(w) = maxL≤|s|≤U

{|s|2|1−w∗a|2 − |s|2

1 + |s|2γ}

= maxL2≤x≤U2

{x|1−w∗a|2 − x

1 + xγ

}, (1.46)

wherex = |s|2. To derive an explicit expression forf(w) we note that the function

h(x) = ax− bx

c + dx(1.47)

with b, c, d > 0 is convex inx ≥ 0. Indeed,

d2h

dx2= 2

bd

(c + dx)3> 0, x ≥ 0. (1.48)

It follows that for fixedw,

g(x) = x|1−w∗a|2 − x

1 + xγ(1.49)

is convex inx ≥ 0, and consequently the maximum ofg(x) over a closed interval is

obtained at one of the boundaries. Thus,

f(w) =

= maxL2≤x≤U2

g(x)

= max(g(L2), g(U2)

)

= max(

L2|1−w∗a|2 − L2

1 + L2γ, U2|1−w∗a|2 − U2

1 + U2γ

),(1.50)


MINIMAX MSE BEAMFORMING FOR KNOWN STEERING VECTOR xxiii

and the problem (1.36) reduces to

minw

{w∗Rw + max

(L2|1−w∗a|2 − L2

1 + L2γ, U2|1−w∗a|2 − U2

1 + U2γ

)}.

(1.51)

We now show that the optimal value ofw has the form

w = d(a∗R−1a)−1R−1a =d

γR−1a, (1.52)

for some (possibly complex)d. To this end, we first note that the objective in (1.51)

depends onw only throughw∗a andw∗Rw. Now, suppose that we are given a

beamformerw, and let

w =a∗wγ

R−1a. (1.53)

Then

w∗a =w∗aγ

a∗R−1a = w∗a, (1.54)

and

w∗Rw =|a∗w|2

γ2a∗R−1a =

|a∗w|2γ

. (1.55)

From the Cauchy-Schwarz inequality we have that for any vectorx,

|a∗x|2 = |(R−1/2a)∗R1/2x|2 ≤ a∗R−1ax∗Rx = γx∗Rx. (1.56)

Substituting (1.56) withx = w into (1.55), we have that

w∗Rw ≤ |a∗w|2γ

≤ w∗Rw. (1.57)

It follows from (1.54) and (1.57) thatw is at least as good asw in the sense of

minimizing (1.51). Therefore, the optimal value ofw satisfies

w =a∗wγ

R−1a, (1.58)

which implies that

w =d

γR−1a, (1.59)

for somed.


xxiv MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

Combining (1.59) and (1.51), our problem reduces to

mind

{ |d|2γ

+ max(

L2|1− d|2 − L2

1 + γL2, U2|1− d|2 − U2

1 + γU2

)}. (1.60)

Sinced is in general complex, we can writed = |d|ejφ for some0 ≤ φ ≤ 2π. Using

the fact that|1 − d|2 = 1 + |d|2 − 2 cos(φ), it is clear that at the optimal solution,

φ = 0. Therefore, without loss of generality, we assume in the sequel thatd ≥ 0.

We can then express the problem of (1.60) as

mint,d

t (1.61)

subject to

d2

γ+ L2(1− d)2 − L2

1 + γL2≤ t;

d2

γ+ U2(1− d)2 − U2

1 + γU2≤ t. (1.62)

The constraints (1.62) can be equivalently written as

fL(d)4=

(1γ

+ L2

) (d− γL2

1 + γL2

)2

≤ t;

fU (d)4=

(1γ

+ U2

)(d− γU2

1 + γU2

)2

≤ t. (1.63)

To develop a solution to (1.61) subject to (1.63), we note that bothfL(d) andfU (d)

are quadratic functions ind, that obtain a minimum atdL anddU respectively, where

dL =γL2

1 + γL2;

dU =γU2

1 + γU2. (1.64)

It therefore follows, that the optimal value ofd, denotedd0, satisfies

dL ≤ d0 ≤ dU . (1.65)

Indeed, lett(d) = max(fL(d), fU (d)), and lett0 = t(d0) be the optimal value of

(1.61) subject to (1.63). Since bothfL(d) andfU (d) are monotonically decreasing

for d < dL, t(d) > t(dL) ≥ t0 for d < dL so thatd0 ≥ dL. Similarly, since both


MINIMAX MSE BEAMFORMING FOR KNOWN STEERING VECTOR xxv

fL(d) andfU (d) are monotonically increasing ford > dU , t(d) > t(dU ) ≥ t0 for

d > dU so thatd ≤ dU .

SincefL(d) andfU (d) are both quadratic, they intersect at most at two points. If

fL(d) = fU (d), then

(1− d)2 =1

(1 + γL2)(1 + γU2), (1.66)

so thatfL(d) = fU (d) for d = d+ andd = d−, where

d± = 1± 1√(1 + γL2)(1 + γU2)

. (1.67)

Denoting byI the intervalI = [dL, dU ], sinced+ > 1, clearlyd+ /∈ I. Using the

fact that1

1 + γU2≤ 1√

(1 + γL2)(1 + γU2)≤ 1

1 + γL2, (1.68)

we have thatd− ∈ I.

In Fig. 1.6, we illustrate schematically the functionsfL(d) andfU (d), where

de = d− = 1− 1√(1 + γL2)(1 + γU2)

(1.69)

is the unique intersection point offL(d) andfU (d) in I. For the specific choices

of fL(d) and fU (d) drawn in the figure, it can be seen that the optimal value of

d is d0 = de. We now show that this conclusion holds true for any choice of

the parameters. Indeed, ifL = U , then de = dL = dU so that from (1.65),

d0 = de. Next, assume thatL < U . In this case, ford ∈ I, fL(d) is monotonically

increasing andfU (d) is monotonically decreasing. Denotingte = t(de) and noting

thatte = fL(de) = fU (de), we conclude that forde < d ≤ dL, fU (d) > te, and for

dU ≤ d < de, fL(d) > te so thatt(d) > te for anyd ∈ I such thatd 6= de, and

therefored0 = de.

One advantage of the minimax regret beamformer is that it explicitly accounts for

both the upper and the lower bounds on|s|, while the minimax MSE beamformer

depends only on the upper bound. Therefore, in applications in which both bounds

are available, the minimax regret beamformer can lead to better performance. As an

example, in Fig. 1.7 we compare the square-root of the NMSE of the minimax regret


xxvi MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

dL de dU

Fig. 1.6 Illustration of the functionsfL(d) andfU (d) of (1.63).

(MXR) and minimax MSE (MXM) beamformers, where for comparison we also plot

the NMSE of the MVDR beamformer, and the MMSE beamformer. Note that the

MMSE beamformer cannot be implemented if|s| is not known, however it serves

as a bound on the NMSE. Each result was obtained by averaging 200 Monte Carlo

simulations. The scenario we consider consists of a ULA ofM = 20 omnidirectional

sensors spaced half a wavelength apart. The signal of interests is a complex random

process, temporally white, whose amplitude has a uniform distribution between the

values 3 and 6 and its plane-wave has a DOA of30o relative to the array normal.

The noisee consists of a zero-mean complex Gaussian random vector, spatially and

temporally white, with a varying power to obtain the desired SNR. The interference

is given byi = aii wherei is a zero-mean complex Gaussian random process with

INR = 20 dB andai its steering vector with DOA =−30o. We assume thatR, a and

the boundsU = 6 andL = 3 are known.

As we expect, the minimax regret beamformer outperforms the minimax MSE and

MVDR beamformers in all the illustrated SNR range, and approaches the performance


RANDOM STEERING VECTOR xxvii

−10 −9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6 −5.5 −50.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75S

quar

e ro

ot o

f NM

SE

SNR [dB]

MMSEMXR MXM MVDR

Fig. 1.7 Square-root of the normalized MSE as a function of SNR using the MMSE, MXR,

MXM and MVDR beamformers in estimating a complex random process with amplitude

uniformly distributed between 3 and 6 and DOA =30o, in the presence of an interference with

INR = 20 dB and DOA =−30o.

of the MMSE beamformer. In Section 1.6, Example 1, we show the advantages of

the minimax MSE and minimax regret beamformers over several other beamformers

for the case wherea is known butR, U andL are estimated from training data

containings.

1.4 RANDOM STEERING VECTOR

In Section 1.3, we explicitly assumed that the steering vector was deterministic and

known. However, it is well known that the performance of adaptive beamformers

degrades due to uncertainties or errors in the assumed array steering vectors [38,

28, 39, 40]. Several authors have considered this problem by modeling the steering

vector as a random vector with known distribution [13] or known second-order

statistics [26, 27, 28, 29, 30, 31, 32, 16]. In this section, we consider the case in

which the steering vector is a random vector with meanm and covariance matrix

C. In this case, the meanm corresponds to a perfectly calibrated array,i.e., the


xxviii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

perturbation-free steering vector, andC represents the perturbations to the steering

vector.

In the case of a random steering vector, the SINR is given by [16, 35]

SINR ∝ w∗Rsww∗Rw

, (1.70)

where

Rs = E {aa∗} = C + mm∗ (1.71)

is the signal correlation matrix, whose rank can be between1 andM . In the case in

which a is deterministic so thata = m, Rs = aa∗ and the SINR of (1.70) reduces

to the SINR of (1.3).

As in the case of deterministica, the most common approach to designing a

beamformer is to maximize the SINR, which results in the principle eigenvector

beamformer

w = αP {R−1Rs

}, (1.72)

whereα is chosen such thatw∗Rsw = 1. The beamformer (1.72) can also be

obtained as the solution to

minw

w∗Rw subject to w∗Rsw = 1. (1.73)

In practice, the interference-plus-noise matrixR is replaced by an estimate.

However, choosingw to maximize the SINR does not necessarily result in an

estimated signal amplitudes that is close tos. Instead, it would be desirable to

minimize the MSE. In the case of a random steering vector, the MSE betweens and

s is given by the expectation of the MSE of (1.13) with respect toa, so that

E{|s− s|2} = w∗Rw + |s|2E {|1−w∗a|2}

= w∗Rw + |s|2E {|1−w∗m−w∗(a−m)|2}

= w∗Rw + |s|2 (|1−w∗m|2 + w∗Cw). (1.74)


RANDOM STEERING VECTOR xxix

The MMSE beamformer minimizing the MSE when|s| is known is obtained by

differentiating (1.74) with respect tow and equating to0, which results in [23]

w(s) = |s|2(R + |s|2C + |s|2mm∗)−1m

=|s|2

1 + |s|2m∗ (R + |s|2C)−1 m

(R + |s|2C)−1

m. (1.75)

Note, that ifC = 0, so thata = m (with probability one), then (1.75) reduces to

w(s) =|s|2

1 + |s|2m∗R−1mR−1m, (1.76)

which is equal to the MMSE beamformer of (1.16) witha = m. Comparing (1.75)

with (1.72) we see that in general the MMSE beamformer and the SINR beamformer

are not scaled versions of each other, as in the known steering vector case. Substituting

w(s) back into (1.74), the smallest possible MSE, which we denote byMSEOPT, is

given by

MSEOPT = |s|2 − |s|4m∗ (R + |s|2C)−1

m. (1.77)

Since the optimal beamformer (1.76) depends explicitly on|s|, it cannot be im-

plemented if|s| is not known. Following the same framework as in the determin-

istic steering vector case, we consider two alternative approaches for designing a

beamformer when|s| is not known: a minimax MSE approach that minimizes the

worst-case MSE over all|s| ≤ U , and a minimax MSE regret that minimizes the

worst-case regret, which in the case of a random steering vector is given by

R(s,w) = w∗Rw + |s|2 (−w∗m−m∗w + w∗(C + mm∗)w)

+ |s|4m∗ (R + |s|2C)−1

m. (1.78)

The minimax MSE and minimax regret estimators for the linear estimation prob-

lem of estimatings in the modely = hs +n, whereh is a random vector with mean

m and covarianceC, andn is a noise vector with covarianceR, has been considered

in [23]. From the results in [23], the minimax MSE beamformer is

wMXMR =U2

1 + U2m∗ (R + U2C)−1 m

(R + U2C

)−1m, (1.79)


xxx MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

which is just an MMSE estimator matched to|s| = U . The minimax regret beam-

former for this problem is

wMXRR =γ

1 + γm∗ (R + γC)−1 m(R + γC)−1 m, (1.80)

whereL2 ≤ γ ≤ U2 is the unique root ofG(γ) defined by

G(γ) =n∑

i=1

|yi|2(U2 − L2

)

(1 + U2λi)(1 + L2λi)λi

((1 + U2λi)(1 + L2λi)

(1 + γλi)2− 1

). (1.81)

Hereyi is the ith component ofy = V∗R−1/2m, V is the unitary matrix in the

eigendecomposition ofA = R−1/2(C + mm∗)R−1/2, andλi is theith eigenvalue

of A.

We see that in the case of a random steering vector, the minimax beamformers

are in general no longer scaled versions of the SINR-based principle eigenvector

beamformer (1.72), but rather point in different directions. Through several numerical

examples (see Section 1.6, Example 2) we demonstrate the advantages of the minimax

MSE and minimax regret beamformers over the principal eigenvector solution as well

as over some alternative robust beamformers [14, 15, 16] for a wide range of SNR

values.

1.4.1 Least-Squares Approach

In the far-field point source case, we have seen that the MVDR approach, which

consists of maximizing the SINR, is equivalent to minimizing the MSE subject to

the constraint that the beamformer is unbiased. The MVDR beamformer is also the

least-squares beamformer, which minimizes the weighted least-squares error

εLS = (y − as)R−1(y − as). (1.82)

As we now show, these equivalences no longer hold in the case of a random steering

vector. First, we note that in this case of a random steering vector, the variance

also depends on the unknown power|s|2. Therefore, in this case, the unbiased

beamformer that minimizes the variance depends explicitly ons and therefore cannot

be implemented. Furthermore, for a full-rank model in whichC is positive definite,


RANDOM STEERING VECTOR xxxi

there is no choice of beamformer that will result in an MSE that is independent ofs

(unless, of course,s = 0). This follows from the fact that the term depending ons in

the MSE is

|s|2 (|1−w∗m|2 + w∗Cw). (1.83)

Sincew∗Cw > 0 for any nonzerow, this term cannot be equal to zero.

Following the ideas in [23], we now consider the least-squares beamformer for the

case of a random steering vector. In this case, the least-squares error (1.82) depends

on a, which is random. Thus, instead of minimizing the errorεLS directly, we may

consider minimizing the expectation ofεLS with respect toa, which is given by

E {εLS} = E{(y −ms− (a−m)s)∗R−1(y −ms− (a−m)s)

}

= (y −ms)∗R−1(y −ms) + s2E{(a−m)∗R−1(a−m)

}

= (y −ms)∗R−1(y −ms) + s2Tr(R−1C

). (1.84)

Differentiating (1.84) with respect tos and equating to0, we have that

s =1

Tr (R−1C) + m∗R−1mm∗R−1y. (1.85)

Thus, the least-squares beamformer is

wLS =1

Tr (R−1C) + m∗R−1mR−1m, (1.86)

which, in general, is different than the MVDR beamformer of (1.72). It is interesting

to note that the beamformer of (1.86) is a scaled version of the MVDR beamformer

for knowna = m.

The advantage of the least-squares approach is that it does not require bounds on the

signal magnitude; in fact, it assumes the same knowledge as the principle eigenvector

beamformer which maximizes the SINR. In Section 1.6 Example 2 we illustrate

through numerical examples that for a wide range of SNR values the least-squares

beamformer has smaller NMSE than the principal eigenvector beamformer (1.72) as

well as the robust solutions [14, 15, 16]. These observations are true even whenm

andC are not known exactly, but are rather chosen in an ad-hoc manner. Therefore,

in terms of NMSE, the least-squares approach appears to often be preferable over


xxxii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

standard and robust methods, while requiring the same prior knowledge. As we

show, the NMSE performance can be improved further by using the minimax MSE

and minimax regret methods; however, these methods require prior estimates of the

signal magnitude bounds.

1.5 PRACTICAL CONSIDERATIONS

In our development of the minimax MSE and minimax regret beamformers, we

assumed that there exists an upper boundU on the magnitude of the signal to be

estimated, as well as a lower boundL for the minimax regret beamformer. For random

steering vectors we assumed also the knowledge of their meanm and covarianceC.

In some applications, the boundsU andL may be known, for example based on

a-priori knowledge on the type of the source and its possible range of distances from

the array. If no such bounds are available, then we may estimate them from the data

using one of the conventional beamformers. Specifically, letwc denote one of the

conventional beamformers. Then, using this beamformer we can estimates(t) as

s(t) = w∗cy(t) (the dependence on the time indext is presented for clarity). We may

then use this estimate to obtain approximate values forU andL. In the simulations

below, we useU = (1 + β)2‖w∗cY‖ andL = (1 − β)2‖w∗

cY‖ for someβ, where

Y is the training data matrix of dimensionM ×N and‖ · ‖ is the average norm over

the training interval.

Since in most applications the true covarianceR is not available we have to

estimate it, e.g., using (1.7). However, as we discussed in Section 1.2, ifs(t) is

present in the training data, then a diagonal loading (1.8) may perform better than

(1.7). Therefore, in the simulations, the trueR is replaced by (1.8).

In the case of a random steering vector, our beamformers rely on knowledge of

the steering vectorm and covarianceC. These parameters can either be estimated

from observations of the steering vector, or, they can be approximated by choosing

m to be equal to a nominal steering vector, and choosingC = νI, whereν reflects

the uncertainty size around the nominal steering vector.


NUMERICAL EXAMPLES xxxiii

1.6 NUMERICAL EXAMPLES

To evaluate and compare the performance of our methods with other techniques,

we conducted numerical examples using scenarios similar to [16]. Specifically, we

consider a uniform linear array ofM = 20 omnidirectional sensors spaced half a

wavelength apart. In all the examples below, we chooses(t) to be either a complex

sinewave or a zero-mean complex Gaussian random process, temporally white, with

varying amplitude or variance respectively, to obtain the desired SNR in each sensor.

The signals(t) is continuously present throughout the training data. The noisee(t)

is a zero-mean, Gaussian, complex random vector, temporally and spatially white,

with a power of 0 dB in each sensor. The interference is given byi(t) = aii(t)

wherei(t) is a zero-mean, Gaussian, complex process temporally white andai is

the interference steering vector. To illustrate the performance of the beamformers

we use the square-root of the NMSE, which is obtained by averaging 200 Monte

Carlo simulations. Unless otherwise stated, we use a signal with SNR=-5 dB, and an

interference with DOA =−30o, INR = 20 dB andN = 50 training snapshots. For

brevity, in the remainder of this section we use the following notation for the different

beamformers: MXM (minimax MSE), MXR (minimax regret), MXMR (minimax

MSE for randoma), MXRR (minimax regret for randoma), LSR (least-squares for

randoma); see also Table 1.1.

We consider two examples: In Example 1 we evaluate the performance of the

MXM and MXR beamformers using the exact knowledge of the steering vector. In

Example 2 we evaluate the MXMR, MXRR and LSR beamformers for a mismatch in

the signal DOA. We focus on low SNR values (important e.g. in sonar) and compare

the performance of the proposed methods against seven alternative methods: the

Capon beamformer (CAPON) [33, 34], loading Capon beamformer (L-CAPON)

[9, 10], eigenspace-based beamformer (EIG) [11, 12], and the robust beamformers

of (1.10), (1.11) and (1.12) which we refer to, respectively, as ROB1, ROB2, and

ROB3 [14, 15, 16]. In Example 2 we compare our methods against the principal

eigenvector beamformer [35], which we refer to as PEIG, withR given by (1.8) and

Rs given by exact knowledge or an ad-hoc estimate (see Example 2 for more details.)


xxxiv MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

Beamformer Expression Parameters Ref.

L-CAPON 1

a∗R−1dl

aR−1

dl a ξ = 10 [10]

EIG 1

a∗R−1eig

aR−1

eiga D = 1 [11]

ROB1 λ

λa∗(Rsm+λε2I

)−1a−1

(Rsm + λε2I

)−1

a ε = 3 [14]

ROB2α1

(λRsm+I

)−1a

√Ma∗

(λRsm+I

)−1Rsm

(λRsm+I

)−1a

ε = 3.5 [15]

ROB3 α2P{R−1

dl (aa∗ − εI)}

ξ = 30, ε = 9 [16]

PEIG α3P{R−1Rs

}[35]

MXM U2

1+U2a∗R−1aR−1a U [41]

MXR α4a∗R−1aR−1a U , L [36]

MXMR U2

1+U2m∗(R+U2C)−1m

(R + U2C

)−1m U , m, C

MXRR γ1+γm∗(R+γC)−1m

(R + γC)−1 m U , L, m, C

LSR 1

Tr(R−1C)+m∗R−1mR−1m m, C

Table 1.1 Beamformers used in the numerical examples.

The parameters of each of the compared methods were chosen as suggested in the

literature. Namely, for L-CAPON (1.8) and PEIG (1.72) the diagonal loading was

set asξ = 10σ2w [14, 15] with σ2

w being the variance of the noise in each sensor,

assumed to be known (σ2w = 1 in these examples); for the EIG beamformer (1.9)

it was assumed that the low-rank condition and number of interferers are known.

For the alternative robust methods, the parameters were chosen as follows: For

ROB1 (1.10) the upper bound on the steering vector uncertainty was set asε = 3

[14], for ROB2 (1.11)ε = 3.5, and for ROB3 (1.12),ε = 9 and the diagonal

loading was chosen asξ = 30. Table 1.1 summarizes the beamformers implemented

in the simulations, whereα1 =∥∥∥∥(R−1

sm + λI)−1

a∥∥∥∥, α2 is chosen such that the

corresponding beamformer satisfiesw∗(aa∗ − εI)w = 1, α3 is chosen such that

w∗Rsw = 1, andα4 = 1− 1/√

(1 + L2a∗R−1a)(1 + U2a∗R−1a).

Example 1 - Known steering vector: In this example we assume that the steering

vectora is known. We first chooses(t) as a complex sinewave with DOA of its


NUMERICAL EXAMPLES xxxv

Beamformer R wc β

MXM Rdl, ξ = 10 L-Capon,ξ = 10 9

MXR Rdl, ξ = 10 L-Capon,ξ = 10 6

Table 1.2 Specification of the MXM and MXR beamformers in Example 1.

plane-wave equal to30o relative to the array normal. We implemented the MXR

and MXM beamformers with the sample covariance matrix estimated using a loading

factorξ = 10 [14, 15],wc given by the L-CAPON beamformer withξ = 10 andβ set

as 6 and 9, for these beamformers, respectively. The values ofβ selected for MXM

and MXR were those that gave the best performance over a wide range of negative

SNR values. Table 1.2 summaries the parameters chosen for MXM and MXR in this

example.

In Fig. 1.8 we plot the square-root of the NMSE as a function of the SNR using

the MXR, MXM, EIG, L-CAPON, ROB2, and ROB3 beamformers. Since in all

the scenarios considered in this example the NMSE of the CAPON and ROB1

beamformers was out of the illustrated scales, we do not plot the performance of

these methods. It can be seen in Fig. 1.8 that the MXM beamformer has the best

performance for SNR values between -10 to -3 dB, and the MXR beamformer has

the best performance for SNR values between -2 to 2 dB. In Fig. 1.9 we plot the

square-root of the NMSE as a function of the number of training data with SNR=-

5 dB. The performance of the proposed methods as a function of the difference

between the signal and interference DOAs is illustrated in Fig. 1.10. The NMSE

of all the methods remains constant for DOA differences between10o and90o, but

deteriorates for DOA differences close to0o. Despite this, MXR and MXM continue

to outperform the other methods. The performance of ROB2 and ROB3 is out of the

illustrated scale for DOA differences less than3o, because the uncertainty region of

their steering vectors overlaps with that of the interference. Figure 1.11 illustrates the

performance as a function of the signal-to-interference-ratio (SIR) for an SNR of -5

dB. As expected, the NMSE of all the beamformers decreases as the SIR increases,


xxxvi MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

−10 −8 −6 −4 −2 0 2 40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Squ

are

root

of N

MS

E

SNR [dB]

MXR MXM EIG L−CAPONROB2 ROB3

Fig. 1.8 Square-root of the normalized MSE as a function of SNR when estimating a complex

sinewave with knowna and with DOA=30o, using the MXR, MXM, EIG, L-CAPON, ROB2,

and ROB3 beamformers.

40 60 80 100 120 140 160 180 2000.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Number of training data

Squ

are

root

of N

MS

E MXR MXM EIG L−CAPONROB2 ROB3

Fig. 1.9 Square-root of the normalized MSE as a function of the number of training snapshots

when estimating a complex sinewave with knowna and DOA=30o, using the MXR, MXM,

EIG, L-CAPON, ROB2, and ROB3 beamformers.


NUMERICAL EXAMPLES xxxvii

0 1 2 3 4 5 6 7 8 9 10

0.4

0.6

0.8

1

1.2

1.4

1.6

Difference between signal and interference DOAs [degrees]

Squ

are

root

of N

MS

EMXR MXM EIG L−CAPONROB2 ROB3

Fig. 1.10 Square-root of the normalized MSE as a function of the difference between the sig-

nal and interference DOAs when estimating a complex sinewave with knowna and DOA=30o,

using the MXR, MXM, EIG, L-CAPON, ROB2, and ROB3 beamformers.

however the minimax methods still outperform the alternative methods for all SIR

values shown.

We next repeat the simulations fors(t) chosen to be a zero-mean complex Gaussian

random signal, temporally white. The square-root of the NMSE as a function of the

SNR, the number of training data, the difference between signal and interference

DOAs, and the SIR is depicted in Figs. 1.12, 1.13, 1.14 and 1.15, respectively. It can

be seen in Fig. 1.12 that the MXM has the best performance for SNR values between

-10 to -4 dB and the MXR has the best performance for SNR values between -3

to 1.5 dB. The performance in Figs. 1.13, 1.14 and 1.15 is similar to the case of a

deterministic sinewave.

Example 2 - Steering vector with signal DOA uncertainties: We illustrate the

robustness of the algorithms for random steering vectors developed in Section 1.4

against a mismatch in the assumed signal DOA. Specifically, the DOA of the signal is

given by a Gaussian random variable with mean equal to30o and standard deviation

equal to 1 (about±3o). This DOA was independently drawn in each simulation run.

To estimate the signal in this case, we implemented the MXRR, MXMR, LSR, and


xxxviii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 00.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

SIR [dB]

Squ

are

root

of N

MS

E MXR MXM EIG L−CAPONROB2 ROB3

Fig. 1.11 Square-root of the normalized MSE as a function of SIR when estimating a complex

sinewave with knowna and DOA=30o, using the MXR, MXM, EIG, L-CAPON, ROB2, and

ROB3 beamformers.

−10 −8 −6 −4 −2 0 2 40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Squ

are

root

of N

MS

E

SNR [dB]


Fig. 1.12 Square-root of the normalized MSE as a function of SNR when estimating a zero-

mean complex Gaussian random signal with knowna and DOA=30o, using the MXR, MXM,



NUMERICAL EXAMPLES xxxix

40 60 80 100 120 140 160 180 200

0.4

0.45

0.5

0.55

0.6

0.65

0.7


Squ

are

root

of N

MS

E


Fig. 1.13 Square-root of the normalized MSE as a function of the number of training snap-

shots when estimating a zero-mean complex Gaussian random signal with knowna and

DOA=30o, using the MXR, MXM, EIG, L-CAPON, ROB2, and ROB3 beamformers.

0 1 2 3 4 5 6 7 8 9 10

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Squ

are

root

of N

MS

E


Fig. 1.14 Square-root of the normalized MSE as a function of the difference between the

signal and interference DOAs when estimating a zero-mean complex Gaussian random signal

with knowna and DOA=30o, using the MXR, MXM, EIG, L-CAPON, ROB2, and ROB3

beamformers.


xl MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 00.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

SIR [dB]

Squ

are

root

of N

MS

E


Fig. 1.15 Square-root of the normalized MSE as a function of SIR when estimating a zero-

mean complex Gaussian random signal with knowna and DOA=30o, using the MXR, MXM,


Beamformer R wc β

MXMR Rdl, ξ = 10 L-Capon,ξ = 10 9

MXRR Rdl, ξ = 10 L-Capon,ξ = 10 4

LSR Rdl, ξ = 10 — —

PEIG Rdl, ξ = 10 — —

Table 1.3 Specification of the beamformers used in Example 2.

PEIG beamformers with parameters given by Table 1.3, for two choices ofm andC:

the true values (estimated from 2000 realizations of the steering vector), and ad-hoc

values. The ad-hoc value ofm was chosen as the steering vector with DOA =30o

and for the covariance we choseC = νI with ν = ε/M , whereε = 3.5 is the norm

value of the steering vector error (size of the uncertainty) used by ROB2 [15] and

M = 20 is the number of sensors. For the ROB1, ROB2 and ROB3 beamformersa

is the steering vector for a signal DOA =30o.


NUMERICAL EXAMPLES xli

−10 −5 0 50

0.5

1

1.5

2

2.5

3

3.5S

quar

e ro

ot o

f NM

SE

SNR [dB]

MXRRMXMRLS PEIGROB1ROB2ROB3

Fig. 1.16 Square-root of the normalized MSE as a function of SNR when estimating a

complex sinewave with random DOA and knownm andC, using the MXRR, MXMR, LSR,

PEIG, ROB1, ROB2, and ROB3 beamformers.

In Figs. 1.16 and 1.17 we depict the NMSE when estimating a complex sinewave

as a function of SNR, using the MXRR, MXMR, LSR, PEIG, ROB1, ROB2, and

ROB3 beamformers, with known and ad-hoc values ofm andC, respectively. As

can be seen from the figures, the MXMR, MXRR and LSR methods perform better

than all the other methods both in the case of knownm andC and when ad-hoc

values are chosen. Although the performance of the MXMR, MXRR and LSR

methods deteriorates when the ad-hoc values are used in place of the true values, the

difference in performance is minor. A surprising observation from the figures is that

the standard PEIG method performs better, in terms of NMSE, than ROB1, ROB2

and ROB3. We also note that the LSR method, which does not use any additional

parameters such as magnitude bounds, outperforms all previously proposed methods.

The performance of the MXRR, MXMR, LSR, PEIG, ROB1, ROB2, and ROB3

beamformers, assuming knownm andC, as a function of the number of training

snapshots and the difference between the signal and interference DOAs is illustrated

in Figs. 1.18 and 1.19, respectively. The NMSE of all methods remain almost

constant for differences above6o. Below this range all the beamformers decrease

their performance significantly, except for the PEIG and LSR beamformers. As in


xlii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

−10 −5 0 50

0.5

1

1.5

2

2.5

3

3.5

Squ

are

root

of N

MS

E

SNR [dB]


Fig. 1.17 Square-root of the normalized MSE as a function of SNR when estimating a

complex sinewave with random DOA and ad-hoc values ofm andC, using the MXRR,

MXMR, LSR, PEIG, ROB1, ROB2, and ROB3 beamformers.

40 60 80 100 120 140 160 180 200

0.5

1

1.5

2

2.5

Squ

are

root

of N

MS

E



Fig. 1.18 Square-root of the normalized MSE as a function of the number of training snap-

shots when estimating a complex sinewave with random DOA and knownm andC using the

MXRR, MXMR, LSR, PEIG, ROB1, ROB2, and ROB3 beamformers.


SUMMARY xliii

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


Squ

are

root

of N

MS

EMXRRMXMRLS PEIGROB1ROB2ROB3

Fig. 1.19 Square-root of the normalized MSE as a function of the difference between the sig-

nal and interference DOAs when estimating a complex sinewave with random DOA and known

m andC using the MXRR, MXMR, LSR, PEIG, ROB1, ROB2, and ROB3 beamformers.

the case of deterministica, the performance of all the methods improves slightly as

a function of negative SIR, and is therefore not shown.

In Figs. 1.20 and 1.21 we plot the NMSE as a function of the SNR when esti-

mating a complex Gaussian random signal, temporally white, with random DOA, as

a function of SNR. As can be seen by comparing these figures with Figs. 1.16 and

1.17, the performance of all of the methods is similar to the case of a deterministic

sinewave. The NMSE as a function of the number of training snapshots and the dif-

ference between the signal and interference DOAs is also similar to the deterministic

sinewave case, and therefore not shown.

1.7 SUMMARY

We considered the problem of designing linear beamformers to estimate a source

signals(t) from sensor array observations, where the goal is to obtain an estimate

s(t) that is close tos(t). Although standard beamforming approaches are aimed

at maximizing the SINR, maximizing SINR does not necessarily guarantee a small


xliv MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION

−10 −8 −6 −4 −2 0 2 40

0.5

1

1.5

2

2.5

3

3.5

SNR [dB]

Squ

are

root

of N

MS

E MXRRMXMRLS PEIGROB1ROB2ROB3


mean complex Gaussian random signal with random DOA and knownm andC, using the

MXRR, MXMR, LSR, PEIG, ROB1, ROB2, and ROB3 beamformers.

−10 −8 −6 −4 −2 0 2 40.5

1

1.5

2

2.5

3

SNR [dB]

Squ

are

root

of N

MS

E



mean complex Gaussian random signal with random DOA and ad-hoc values ofm andC,

using the MXRR, MXMR, LSR, PEIG, ROB1, ROB2, and ROB3 beamformers.


SUMMARY xlv

MSE, hence on average a signal estimate maximizing the SINR can be far froms(t).

To ensure thats(t) is close tos(t), we proposed using the more appropriate design cri-

terion of MSE. Since the MSE depends in general ons(t) which is unknown, it cannot

be minimized directly. Instead, we suggested beamforming methods that minimize

a worst-case measure of MSE assuming known and random steering vectors with

known second-order statistics. We first considered a minimax MSE beamformer that

minimizes the worst-case MSE. We then considered a minimax regret beamformer

that minimizes the worst-case difference between the MSE using a beamformer ig-

norant ofs(t) and the smallest possible MSE attainable with a beamformer that

knowss(t). As we showed, even ifs(t) is known, we cannot achieve a zero MSE

with a linear estimator. In the case of a random steering vector we also proposed a

least-squares beamformer that does not require bounds on the signal magnitude.

In the numerical examples, we clearly illustrated the advantages of our methods

in terms of the MSE. For both known and random steering vectors, the minimax

beamformers consistently have the best performance, particularly for negative SNR

values. Quite surprisingly, it was observed that the least-squares beamformer, which

does not require bounds on the signal magnitude, performs better than the recently

proposed robust methods in the case of random DOAs. It was also observed that

for a small difference between signal and interference directions of arrival, all our

methods show better performance. The performance of our methods was similar

when the signal was chosen as a deterministic sinewave or a zero-mean complex

Gaussian random signal.

Acknowledgments

The authors are grateful to Mr. Patricio S. La Rosa for conducting the numerical examples

and for carefully reviewing several earlier versions of this chapter. The work of A. Nehorai

was supported by the Air Force Office of Scientific Research Grant F49620-02-1-0339, and

the National Science Foundation Grants CCR-0105334 and CCR-0330342.


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Index

Bias, viii, xv

Capon beamformer, vi, xxxiii

Complex sinewave, x, xxxiii–xxxiv, xli

Correlation, xxviii

Covariance, v–vii, xi–xii

estimated, vi

interference-plus-noise, i, vi–vii, xvi

sample, i, vi, xxxv

Diagonal loading, vi, xxxii, xxxiv

DOA, x, xii, xxvi, xxxiii–xxxv

Eigenspace, vii, xxxiii

Estimation, i–ii, viii, x, xxix

error, ii, viii

Expectation, xxviii, xxxi

Least-favorable, xxi

Least-squares, iii, xxx–xxxi, xxxiii, xlv

Loading Capon beamformer, xxxiii

Lower bound, iii, xv, xvii–xviii, xxv, xxxii

Mean, iii, v, xi–xii, xx–xxii, xxvii, xxix, xxxii,

xxxvii

Minimax, ii

beamformer, iii, xxx, xlv

MSE, ii–iv, ix, xvi–xviii, xx, xxii, xxv–xxvii,

xxix–xxx, xxxii–xxxiii, xlv

regret, ii–iv, xvi, xviii, xx–xxi, xxv–xxvii,


MMSE, ix–xiii, xvi–xviii, xxi, xxvi–xxvii,

xxix–xxx, liSEMSE

MSE, ii–iv, viii–xi, xiii, xv–xxii, xxv–xxxiii, xlv

MVDR, vi–xi, xv–xvii, xx, xxvi, xxx–xxxi

NMSE, x, xii, xxv–xxvi, xxxi–xxxiii, xxxv,

xxxvii, xli, xliii

Overconservative, xviii

Principle eigenvector, xxviii, xxx–xxxi

Regret, ii–iv, ix, xvi, xviii–xxii, xxv–xxvii,


Robust beamformer, ii, iv, vii–viii, xv–xvi, xxx,

xxxiii

Shrinkage, x

Signal amplitude, ii, iv–v, viii, x, xii, xx, xxviii

SINR, i–iii, v–xii, xv, xxviii–xxxi, xliii, xlv

SIR, xxxv, xxxvii, xliii

SNR, i, iii–iv, x–xiv, xxi, xxvi, xxx–xxxi, xxxiii,

xxxv, xxxvii, xli, xliii

Steering vector, i–v, vii–viii, x–xvi, xxvi–xxxv,

xxxvii, xl, xlv, li

nominal, v, xxxii

D R A F T September 23,2004, 8:12pm D R A F T

INDEX liii

random, ii–iv, x–xii, xv–xvi, xxvii–xxxii,

xxxvii, xlv

Training data, i, vi–vii, xxvii, xxxii–xxxiii, xxxv,

xxxvii

ULA, x, xxvi

Unbiased, xv, xxx

Uncertainty, i–iii, v, vii, xvi, xviii, xxi, xxxii,

xxxiv–xxxv, xl

Upper bound, xv–xvi, xviii, xxv, xxxii, xxxiv

Variance, iii, vi, viii–ix, xxx, xxxiv

Worst-case, xvi

MSE, ii, xvi, xix, xxix, xlv

regret, iii, xviii


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